Despite the somewhat esoteric nature of Carathéodory’s axiomatic approach to thermodynamics via the geometric behavior of Pfaffians – or perhaps even because of it – my blogpost “Carathéodory: the forgotten pioneer” has received a surprisingly large number of hits, with plenty of brave individuals willing to click on the link to the English version of Carathéodory’s original paper published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics].

Carathéodory’s second axiom “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes”, and the associated theorem giving the condition for dQ to be an integrable differential, constitute the real novelty of his approach.

My original post described Carathéodory’s theorem without going into the proof, since it is rather abstruse and would have appealed only to more avid students of his work. Two years on however, the statistics for this blogpost reveal that there are plenty of avid students wanting to know more. So as a supplementary post, here is a proof of Carathéodory’s theorem, due to Pierre Perrot. Enjoy.

– – – –

Carathéodory’s theorem

“If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.”

In the following, use is made of a classical result known as the Clausius inequality

Proof

Cases n = 1 and n = 2 are trivial because a differential function of only one variable is necessarily total whereas a differential function of two variables is necessarily integrable. All points accessible to a given point P form a continuous domain around P. In an n-dimensional space (n≥3) around P, this domain fills a volume [n dimensions], or a surface [(n-l) dimensions], or a curve [≤ (n-2) dimensions].

The first possibility is excluded because it contradicts the hypothesis that around P there are points which are inaccessible. The third possibility is also excluded because the expression dQ = 0 already defines a surface element containing only points accessible to P. Therefore, points close to P and accessible to P define only a surface. If we now consider a point P’ on that surface, it is impossible to go from P to P’ by a curve satisfying the condition ∫dQ = 0 and not situated on this surface, otherwise every point situated within the immediate proximity of P would be accessible, which contradicts the hypothesis.

From a point P1 it is possible to define a surface S1, upon which all points are accessible to P1. Also, from a point P2 not situated on S1, it is possible to define a surface S2. Surfaces S1 and S2 have no common point between them, otherwise it would be possible to go from P1 to P2 by a path such that ∫dQ = 0. Therefore there is a family of surfaces where σ(x1, x2,…, xn) is constant, filling the space and having no common point among them. For this one-parameter family, dσ = 0 implies dQ = 0, from which, between dQ and dσ, there exists a relation of the type:

where, because dQ = ΣXidxi

Naturally, the family of surfaces for which σ is constant may also be expressed by S(σ) = constant, where S(σ) is an arbitrary function of σ:

Hence

(l/T) is the integrating factor. If a differential dQ has one integrating factor, it has an infinity, S being an arbitrary function of σ.

– – – –

Summary

Carathéodory’s theorem shows that if a differential dQ is integrable, the equation dQ = 0 characterizes in a space a family of surfaces sharing no common point. For any point P on one of these surfaces, it is always possible to find, immediately near that point, points which do not belong to the surface and which are therefore inaccessible by a curve solution of the equation dQ = 0. On the other hand, if dQ is not integrable, the equation dQ = 0 does not define any surface in the space and it will always be possible to link any two points with a curve solution of the equation dQ = 0.